Group #34000471 — subgroups

Subgroups — #34000471

Order 18688.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625810	C₂	1	2	yes	maximal
2	#22611541	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556661	—	2	2	no	—
4	#17989558	—	2	2	no	—
5	#17989463	C₄	1	4	yes	—
6	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
7	C₂×#14556531	C₄	1	1	yes	characteristic
8	#17988366	—	2	2	no	—
9	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924038	—	2	4	no	—
11	#14556661	—	4	2	no	—
12	#14556661	—	4	2	no	—
13	#14556648	—	4	2	no	—
14	C₂²⋊#10596833	—	4	2	no	—
15	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
16	C₂²×#10596831	Q₈	1	1	yes	characteristic
17	C₂×C₈×D₇₃	—	4	1	no	—
18	C₂×C₈×D₇₃	—	4	1	no	—
19	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₇₃	D₄	1	2	yes	—
21	#14556531	—	4	1	no	—
22	C₂×#11924003	—	2	1	no	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	M₃₂×C₇₃	—	4	1	no	—
25	C₂×C₁₆×C₇₃	—	2	1	no	—
26	#11924038	—	4	4	no	—
27	#11924038	—	4	4	no	—
28	C₂×#10596833	—	4	4	no	—
29	C₂³×D₇₃	Q₁₆	1	1	yes	characteristic
30	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
31	C₂×C₄×D₇₃	—	4	1	no	—
32	C₂×C₄×D₇₃	—	4	1	no	—
33	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
35	C₂×#10596831	—	4	1	no	—
36	C₂×#10596831	C₄⋊C₄	1	2	yes	—
37	C₈×D₇₃	—	2	2	no	—
38	C₈×D₇₃	—	2	2	no	—
39	C₈×D₇₃	—	4	2	no	—
40	#11924003	—	4	1	no	—
41	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	#107	1	1	yes	characteristic
48	C₂²×D₇₃	—	4	1	no	—
49	C₂²×D₇₃	—	4	2	no	—
50	C₂²×D₇₃	#103	1	2	yes	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	#102	1	2	yes	—
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	2	2	no	—
59	C₄×D₇₃	—	4	2	no	—
60	#10596833	—	8	4	no	—
61	C₈×C₇₃	—	2	1	no	—
62	C₈×C₇₃	#99	1	1	yes	characteristic
63	C₈×C₇₃	#99	1	1	yes	characteristic
64	#10596831	—	2	2	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	#340	1	1	yes	characteristic
68	C₂×D₇₃	—	2	2	no	—
69	C₂×D₇₃	—	2	4	no	—
70	C₂×D₇₃	—	4	4	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
73	C₄×C₇₃	#342	1	1	yes	characteristic
74	Dic₇₃	—	2	2	no	—
75	#7234	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	4	1	no	—
77	C₂×C₇₃	#1289	1	1	yes	characteristic
78	C₂×C₇₃	#1388	1	2	yes	—
79	D₇₃	—	4	2	no	—
80	#2112	—	73	1	no	—
81	#1810	—	73	2	no	—
82	C₇₃	#7234	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₈	—	73	1	no	—
84	C₂×M₃₂	—	73	1	no	—
85	C₂×M₃₂	#64408	1	1	yes	characteristic
86	C₂×#108	—	146	2	no	—
87	#407	—	146	2	no	—
88	C₂²⋊C₁₆	—	146	2	no	—
89	#341	—	73	4	no	—
90	C₂³×C₄	—	73	1	no	—
91	C₂×M₁₆	—	146	4	no	—
92	C₂²×C₈	—	292	1	no	—
93	C₂²×C₈	—	292	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	2	no	—
96	C₂²×C₈	—	73	2	no	—
97	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
98	M₃₂	—	4	1	no	—
99	M₃₂	—	292	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	146	1	no	—
102	#108	—	292	2	no	—
103	#108	—	292	2	no	—
104	C₂²⋊C₈	—	292	2	no	—
105	C₂²⋊C₈	—	292	2	no	—
106	C₂⁴	—	73	1	no	—
107	C₂²×C₄	—	292	1	no	—
108	C₂²×C₄	—	292	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂²×C₄	—	73	2	no	—
112	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
113	M₁₆	—	292	4	no	—
114	M₁₆	—	292	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	292	2	no	—
118	C₂×C₈	—	292	4	no	—
119	C₂×C₈	—	292	1	no	—
120	C₂×C₈	—	73	2	no	—
121	C₂×C₈	—	146	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	C₄×#64408	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
125	C₁₆	—	4	1	no	—
126	C₁₆	—	292	1	no	—
127	C₂³	—	292	1	no	—
128	C₂³	—	292	2	no	—
129	C₂³	—	73	2	no	—
130	C₂³	#14556660	1	1	yes	characteristic
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	292	2	no	—
134	C₂×C₄	—	292	1	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	2	no	—
138	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	#14556659	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	584	4	no	—
142	C₈	—	146	2	no	—
143	C₈	#14556649	1	1	yes	characteristic
144	C₈	#14556650	1	1	yes	characteristic
145	C₂²	—	2	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	292	4	no	—
148	C₂²	—	146	2	no	—
149	C₂²	—	146	4	no	—
150	C₂²	#17989461	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	146	2	no	—
153	C₄	#17989455	1	1	yes	characteristic
154	C₄	#17989465	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	—	292	2	no	—
157	C₂	#22624794	1	1	yes	characteristic
158	C₂	#22625009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625810

C₂

yes

maximal

#22611541

C₂

yes

maximal, characteristic

C₂×#14556661

—

#17989558

—

#17989463

C₄

yes

—

C₂²×C₈×D₇₃

C₂²

yes

characteristic

C₂×#14556531

C₄

yes

characteristic

#17988366

—

C₂×M₃₂×C₇₃

C₄

yes

—

—

—

—

—

C₂²×C₄×D₇₃

D₄

yes

characteristic

C₂²×#10596831

Q₈

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×C₈×D₇₃

D₄

yes

—

#14556531

—

C₂×#11924003

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#11924038

—

#11924038

—

C₂×#10596833

—

C₂³×D₇₃

Q₁₆

yes

characteristic

C₂²×Dic₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

—

C₂×C₄×D₇₃

SD₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#107

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

#103

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#7234

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1289

yes

characteristic

C₂×C₇₃

#1388

yes

—

D₇₃

—

#2112

—

#1810

—

C₇₃

#7234

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

C₂×#108

—

146

—

#407

—

146

—

C₂²⋊C₁₆

—

146

—

#341

—

C₂³×C₄

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

292

—

C₂²×C₈

—

292

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

M₃₂

—

M₃₂

—

292

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

146

—

102

#108

—

292

—

103

#108

—

292

—

104

C₂²⋊C₈

—

292

—

105

C₂²⋊C₈

—

292

—

106

C₂⁴

—

107

C₂²×C₄

—

292

—

108

C₂²×C₄

—

292

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄⋊#64408

yes

characteristic

113

M₁₆

—

292

—

114

M₁₆

—

292

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

C₄×#64408

yes

characteristic

124

C₂×C₈

C₂²⋊#64408

yes

characteristic

125

C₁₆

—

126

C₁₆

—

292

—

127

C₂³

—

292

—

128

C₂³

—

292

—

129

C₂³

—

130

C₂³

#14556660

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

139

C₂×C₄

#14556659

yes

characteristic

140

C₈

—

141

C₈

—

584

—

142

C₈

—

146

—

143

C₈

#14556649

yes

characteristic

144

C₈

#14556650

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

—

292

—

148

C₂²

—

146

—

149

C₂²

—

146

—

150

C₂²

#17989461

yes

characteristic

151

C₄

—

152

C₄

—

146

—

153

C₄

#17989455

yes

characteristic

154

C₄

#17989465

yes

characteristic

155

C₂

—

156

C₂

—

292

—

157

C₂

#22624794

yes

characteristic

158

C₂

#22625009

yes

—

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×D₇₃	Q₁₆
cosocle	C₂²×C₈×D₇₃	C₂²
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#7234
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂×M₃₂×C₇₃

C₄

socle

C₂²×C₇₃

#340

prime radical

C₂³×D₇₃

Q₁₆

cosocle

C₂²×C₈×D₇₃

C₂²

C₂×M₃₂

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
82. C₇₃	#7234	semidirect

H ≅ G/N

type

82. C₇₃

#7234

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22625810	2
#22611541	2
#7234	73

subgroup

index

#22625810

#22611541

#7234

Subgroups — #34000471

Conjugacy classes of subgroups

Complements

Maximal subgroups